Narrowband array processing beamforming technique for electrical impedance tomography

Elliptic partial differential equations (PDEs) typically characterize boundary value problems with steady state conditions in two dimensions and hence can be employed to characterize the spatial distribution of an electrostatic field potential in EIT systems. The distribution of an electric potential within an object through which a steady current flows, can be represented by the solution of Poisson’s equation, that reduces to Laplace’s equation for a source free, homogeneous medium. The numerical solution for such elliptic PDEs is based on treating the test object as a grid of discrete points or nodes and transforming these PDEs into an algebraic finite difference equations [13].

The interior nodes are represented in terms of the Laplacian difference equations. The boundary conditions must be specified in order to obtain a unique solution. There would be more linear algebraic equations depending on the number of interior nodes. However, there would be a maximum of five (5) unknowns for each node (equation). For a large sized grids, this means that a significant number of terms would be zero. This could lead to the wastage of computer memory in storing zero’s when full matrix elimination methods are used. Hence, approximation methods are used for obtaining the solutions for elliptical equations.

The numerical solution is provided by the most commonly used Gauss-Seidal method which when applied to partial differential equations, is referred to as Liebmann’s method [11]. The matrix obtained is diagonally dominant and hence would converge on to a stable solution. The iterations are repeated until the absolute values of all the percentage relative errors (ε_a) fall below a specified stopping criteria, ε_s (usually 1%). The percentage relative errors are estimated by equation (1). (1)|(εa)|=|V(new)-V(old)V(new)|×100%\left| {\left( {\varepsilon _a } \right)} \right| = \left| {{{V(new) - V(old)} \over {V(new)}}} \right| \times 100\%

The block diagram of the developed beamforming EIT system is shown in Fig. 1. We assume that the currents do not flow in a straight line, instead, follow a path of higher conductivity regions. Also, the contact impedance issues are not considered in the study.

Fig. 1

A recyclable, polyethylene terephthalate (PET) circular container of around 11 cm in diameter and 16 cm in height was used for the proposed study. Eight holes at every 4.4 cm were drilled along the circumference of the container and 6 cm from the bottom of the container. The holes were tightly sealed with the Ag/AgCl electrodes and then saline water (10% NaCl) of 12 cm height was filled in the container. A standard check was carried out by measuring the contact impedances of all the electrodes, which measured 0.6 Ω. Apart from that, the electrode considerations, excitation methods and voltage measurement techniques are according to [14].

CD4067BE is a single 16-channel multiplexer/demultiplexer IC. Only 8 channels were used for the intended study. (although only one IC was sufficient, two ICs were used for the possible extension of the number of electrodes to 16 in the future). These two multiplexers are used to perform the current injection into the test tank through the electrodes. One multiplexer is used for selecting the current source while the other multiplexer would select the ground. The signal from a function generator is injected into MUX I+ while MUX I- is connected to the ground. The function generator (Kikusui Electronics Corp, model 459) is used as a current source for the EIT system. The output current is determined by the output impedance and the output voltage.

With an output resistance of 600Ω and a maximum output voltage of 20 V_PP, a current of up to 33 mA could be generated. A safe let-go current of 6 mA [15] at 50 kHz is chosen as a mid-value from the optimum frequency range for electrical impedance tomography applications which is from 10 kHz to 100 kHz [16] for the proposed study.

The National Instruments, NI USB 6008 DAQ (data acquisition) card acquires the voltage data from the electrodes as well as acts as a control device for the multiplexers. The complete experimental set up of the EIT system using eight (8) electrodes is shown in Fig 2.

Fig. 2

A program is developed to shift the currents according to a sequence using a multiplexer circuit. Two-electrode configuration is used where the input current is applied to a pair of electrodes, which would not measure any voltage readings. Thus, each electrode will have six analog voltages (rms values) for one cycle, as the other two readings will be zero during current injection. The program gets the sum of the individual electrode voltage readings after one complete cycle.

The Laplace equation for a circular test tank can be represented in the polar co-ordinates by equation (2), containing no free charges. (2)∂2V∂r2+∂2V∂θ2=0{{\partial ^2 V} \over {\partial r^2 }} + {{\partial ^2 V} \over {\partial \theta ^2 }} = 0 where, r is the radius of the circle in centimeter and θ is the angle between two consecutive electrodes in radians.

Fig. 3 shows the circular test tank of radius r₁ for the intended EIT study. The test tank is insulated everywhere except at the points, where the electrodes, V₁–V₈, are connected, at an angular difference of 45 degrees. Also, the conductivity is assumed to be uniform inside the test tank.

Fig. 3

After acquiring the boundary electrode potentials, more boundary potentials are interpolated by bi-curvilinear interpolation given by equation (3), which are shown as V_b1–V_b8 in Fig. 3. (3)Vbk(r,θ)=V2Vb1V2V1×V1(r1,θ1)+Vb1V1V2V1×V2(r2,θ2)V_{bk} (r,\theta ) = {{V_2 V_{b1} } \over {V_2 V_1 }} \times V_1 \left( {r_1 ,\theta _1 } \right) + {{V_{b1} V_1 } \over {V_2 V_1 }} \times V_2 \left( {r_2 ,\theta _2 } \right) where, V₂V_b1 and V_b1V₁ are arc lengths, k = 1⋯8, r₁ = r₂ and θ₁ = 22.5 degrees while θ₂ = 45 degrees.

Now for interpolating the internal nodal voltages, the circular test tank is treated as grids of discrete points as shown once again in Fig. 3. The central difference equations based on the grid scheme are given by equation (4) and equation (5). (4)∂2V∂r2=Vi+1,j-2Vi,j+Vi-1,jΔr2{{\partial ^2 V} \over {\partial r^2 }} = {{V_{i + 1,j} - 2V_{i,j} + V_{i - 1,j} } \over {\Delta r^2 }}(5)∂2V∂θ2=Vi,j+1-2Vi,j+Vi,j-1Δθ2{{\partial ^2 V} \over {\partial \theta ^2 }} = {{V_{i,j + 1} - 2V_{i,j} + V_{i,j - 1} } \over {\Delta \theta ^2 }} The radius is taken as the first dimension while the arc length is considered instead of the angle, θ, (equation (6)) as the second dimension. (6)∂2V∂θ2=Vi,j+1-2Vi,j+Vi,j-1Δ∩2{{\partial ^2 V} \over {\partial \theta ^2 }} = {{V_{i,j + 1} - 2V_{i,j} + V_{i,j - 1} } \over {\Delta \cap ^2 }} Substituting equation (4) and equation (6) in equation (2), (7)Vi+1,j-2Vi,j+Vi-1,jΔr2+Vi,j+1-2Vi,j+Vi,j+1Δ∩2=0{{V_{i + 1,j} - 2V_{i,j} + V_{i - 1,j} } \over {\Delta r^2 }} + {{V_{i,j + 1} - 2V_{i,j} + V_{i,j + 1} } \over {\Delta \cap ^2 }} = 0Equation (7) is the Laplacian difference equation for the interior nodal voltages. Assuming, Δr = Δ∩ = h, the matrix structure of the five-point approximation can be represented as, (0101-41010)(*Vi,j+1*Vi-1,jVi,jVi+1,j*Vi,j-1*)=h2\left( {\matrix{0 & 1 & 0 \cr 1 & { - 4} & 1 \cr 0 & 1 & 0 \cr } } \right)\left( {\matrix{* & {V_{i,j + 1} } & * \cr {V_{i - 1,j} } & {V_{i,j} } & {V_{i + 1,j} } \cr * & {V_{i,j - 1} } & * \cr } } \right) = h^2

The boundary potentials must be specified in order to obtain a unique solution. Laplacian difference equations are developed for each interior nodal voltage, which result in an ‘n’, number of simultaneous equations with (n-1) unknowns (V_{(i, j)}, the central voltage is assumed as the average of the measured boundary voltages for a homogeneous saline region). For any other value(s), the electrostatic potential distribution would be misleading). The numerical solution for the above Laplacian difference equations are obtained as discussed in numerical methods section. The rate of convergence can be further accelerated using over-relaxation methods [17].

A time difference approach is used to locate the test objects. Initially, a set of voltage readings is taken of the empty saline tank as a reference frame. Later, a second set of voltage readings are taken with the test object inside the test tank. Finally, the difference of the voltage readings of the two sets is fed as the boundary voltages. After the above interpolations, the concept of narrowband beamformer is applied where all the voltage values are summed up w.r.t. a particular angle. A test object is detected as a peak in the output voltage signal. Finally, suitable weight (a multiplication factor) is chosen independent of the electrode data for the accurate representation of the test object along its radius.

The test tank is filled with saline (10% NaCl) water for better conductivity. An eight electrode EIT system is considered for the simplicity of the design. A current of 6 mA at 50 kHz is injected adjacently w.r.t. ground. In return, the raw mono-polar voltages (assuming the noise levels to be negligible in the measured data, high SNR) are measured from the remaining non-current carrying electrodes in a cyclic manner which are later summed up for successive current patterns in a single measurement cycle. The basic image reconstruction of EIDORS [18] is considered for the comparison of the reconstructed images. A test object in the form of a hollow cylindrical iron rod was used. The dimensions include 15.5 cm length and 2.5 cm diameter. The test object(s) would be placed in line with the electrodes and then in the vicinity of the electrodes.

The conducted research is not related to either human or animal use.

When a single test rod is placed exactly in line with the electrode locations (not touching the electrodes), the conventional EIDORS gives distorted images due to contact impedance issues. On the other hand, the beamforming algorithm shows the exact location of the test object. A peak output in the beamforming represents the location of the test object as shown in Fig. 4. Hence, the beamforming algorithm is good for detection.

Fig. 4

Similar results are visible as compiled in Fig. 5 for different locations of the test object with the same considerations.

Fig. 5

Similarly, when two cylindrical test rods of same material and same dimensions were placed in the test tank, once again exactly in line with the electrode positions, EIDORS had the same contact impedance issues. While, the beamforming algorithm showed the exact locations of the test objects as seen in Fig. 6. Hence, the beamforming is good for distinguishability.

Fig. 6

Similar results are visible as compiled in Fig. 7 for the two test objects in different locations with the same considerations.

Figure 7

However, when a test object is placed between two electrode positions, the beamforming shows a flat response running across the two electrodes instead of showing a peak in between the two electrodes as seen in Fig. 8(a). This is due to the similar voltage drops captured by the two electrodes when the test object is placed between them. The same condition is also visible with two test objects placed between any two electrodes as shown in Fig. 8(b).

Fig. 8

The problem of flat beam response can be solved by using more electrodes in order to indicate the exact location(s) of the test object(s). Even three test objects could be detected exactly by the beamforming algorithm when placed exactly along the electrode alignment. A maximum of four (4) test objects can be detected by the developed EIT system. For the test tank of 11 cm diameter with eight electrodes at every 4.4 cm along the circumference, a decent spatial resolution of 6.3 cm is achieved without a flat beam response.

However, the spatial resolution could be improved further by simply increasing the number of electrodes. For example, for the same test tank with 16 electrodes, the spatial resolution could be 3.15 cm and with 32 electrodes would be 1.575 cm and so on. But, there is a limit for increasing the number of electrodes beyond which the image resolution is no longer improved [19]. More electrodes also mean more hardware and hence more processing time. Moreover, if an EIT system uses a number of electrodes, then all the electrodes must be driven by the input currents simultaneously for better results [20].

All the test results show conductivity beam(s) when the test object(s) are placed in the vicinity of the electrodes (along the periphery). However, when placed in the middle of test tank, the beamforming (only the conductors were tested for the intended study) algorithm showed a flat beam. Moreover, there was no object detected by the system when the test objects were smaller than 1 cm in diameter. The numerical solutions discussed are limited to circular test objects. Hence, prior modeling of any other shapes is required for better images. Also, neither the shape nor the size of the test object can be known.

Therefore, there is place for improvement of the measurement system by the following ways: by decreasing the size and increasing the number of electrodes (at least up to 16), by applying the better electrode setup (e.g. four electrode configuration), in trying other current injection patterns; e.g., some current injection patterns such as opposite or polar driven pattern produce more field strength in deeper regions of the medium and in this way provide images with higher resolution, by using a central electrode for more accurate readings, by employing more advanced image reconstruction algorithms like time-difference algorithms using multi-frequency information [21] and/or deep learning beamforming algorithms [22], by employing numerical solutions which are not limited to circular objects, by considering the noise levels in the measured data due to EIT circuitry and also by considering the contact impedances issues.

All the above suggestions for improvement of the measurement system would contribute to decreasing the contact impedance, increasing the signal to noise ratio and hence increasing the spatial resolution of the EIT images. Thus, it is suggested that these would be taken into consideration for future perspectives in order to enable EIT as a common imaging modality